Adaptive Fractional Higher Interpolatory Cubature Kalman Filter with Online Noise Covariance Estimation
Keywords:
Fractional calculus, fractional nonlinear stochastic systems, higher interpolatory cubature rule, Adaptive processAbstract
Aiming at the challenge of state estimation in the fractional nonlinear discrete stochastic systems, we propose an adaptive fractional higher interpolatory cubature Kalman filter (AFHICKF). We develop the AFHICKF algorithm by using higher-degree interpolatory cubature rules to fulfill the numerical integral computation under the framework of Bayesian fractional filtering. Moreover, the adaptive process of AFHICKF is designed to address the state estimation problem to fractional nonlinear discrete stochastic systems with unknown noise covariance through online covariance estimation. Simulation results on reentry target tracking system verify the effectiveness, adaptiveness and superiority of the proposed filter.
References
[1] Mohamed AH and Schwarz KP. Adaptive Kalman Filtering for INS/GPS. Joural of Geodesy, 1999, p. 193.
https://doi.org/10.1007/s001900050236
[2] Ali Mehrjouyan AA. Robust adaptive unscented Kalman filter for bearings-only tracking in three dimensional case. Applied Ocean Research 2019; 87 223–232.
https://doi.org/10.1016/j.apor.2019.01.034
[3] Zhu H, Zhang GR, Li YF, et al. An Adaptive Kalman Filter With Inaccurate Noise Covariances in the Presence of Outliers. IEEE Trans Automat Contr 2022; 67: 374-381.
https://doi.org/10.1109/TAC.2021.3056343
[4] Qiu ZB and Guo L. Improved Cubature Kalman Filter for Spacecraft Attitude Estimation. IEEE Transactions on Instrumentation and Measurement 2021; 70.
https://doi.org/10.1109/TIM.2020.3041077
[5] Julier S UJ, Durrant-Whyte HF. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Transactions on Automatic Control 2000; 45: 477-482.
https://doi.org/10.1109/9.847726
[6] Simon J. Juler JKU. Unscented filtering and nonlinear estimation. Proceedings of the IEEE,2004. 92(3), p 401-422.
https://doi.org/10.1109/JPROC.2003.823141
[7] Ienkaran Arasaratnam SH. Cubature Kalman Filters. IEEE Transactions on Automatic Control United States 2009, p. 1254-1269.
https://doi.org/10.1109/TAC.2009.2019800
[8] Yonggang Zhang YH, Ning Li, Lin Zhao. Interpolatory cubature Kalman filters. IET Control Theory & Applications 2015; 9: 1731–1739.
https://doi.org/10.1049/iet-cta.2014.0873
[9] Raman K M. Approaches to Adaptive Filtering. IEEE Transactions on Automatic Control 1972; 17: 693-698.
https://doi.org/10.1109/TAC.1972.1100100
[10] Akbar Assa KNP. Adaptive Kalman Filtering by Covariance Sampling. IEEE Signal Processing Letters 2017, 24(9), p1288-1292.
https://doi.org/10.1109/LSP.2017.2724848
[11] Hairong Wang ZD, Bo Feng, Hongbin Ma, Yuanqing Xia. An adaptive Kalman filter estimating process noise covariance. Neurocomputing 2017; 223: 12-17.
https://doi.org/10.1016/j.neucom.2016.10.026
[12] Chang GB, Chen C, Zhang QZ, et al. Variational Bayesian adaptation of process noise covariance matrix in Kalman filtering. Journal of the Franklin Institute-Engineering and Applied Mathematics 2021; 358: 3980-3993.
https://doi.org/10.1016/j.jfranklin.2021.02.037
[13] Akhlaghi S, Zhou N and Huang Z. Adaptive adjustment of noise covariance in Kalman filter for dynamic state estimation. In: 2017 IEEE Power & Energy Society General Meeting 16-20 July 2017 2017, pp.1-5.
https://doi.org/10.1109/PESGM.2017.8273755
[14] Assa A, Janabi-Sharifi F and Plataniotis KN. Sample-based adaptive Kalman filtering for accurate camera pose tracking. Neurocomputing 2019; 333: 307-318.
https://doi.org/10.1016/j.neucom.2018.11.083
[15] De Vivo F, Brandl A, Battipede M, et al. Joseph covariance formula adaptation to Square-Root Sigma-Point Kalman filters. Nonlinear Dynamics 2017; 88: 1987-1987.
https://doi.org/10.1007/s11071-017-3406-4
[16] Zhang Y, Huang Y, Li N, et al. Embedded cubature Kalman filter with adaptive setting of free parameter. Signal Processing 2015; 114: 112-116.
https://doi.org/10.1016/j.sigpro.2015.02.022
[17] Aritro Dey MD, Smita Sadhu, Tapan Kumar Ghoshal. Adaptive divided difference filter for parameter and state estimation of non-linear systems. IET Signal Processing, 2015, p. 369-376.
https://doi.org/10.1049/iet-spr.2013.0395
[18] Jang HN, Cai YL. Adaptive Fifth-Degree Cubature Information Filter for Multi-Sensor Bearings-Only Tracking. sensors 2018.
https://doi.org/10.3390/s18103241
[19] Manasi Das AD, Smita Sadhu, Tapan Kumar Ghoshal. Adaptive central difference filter for non-linear state estima-tion. IET Science, Measurement & Technology 2015; 9: 728-733.
https://doi.org/10.1049/iet-smt.2014.0299
[20] Shan C, Zhou W, Yang Y, et al. Multi-Fading Factor and Updated Monitoring Strategy Adaptive Kalman Filter-Based Variational Bayesian. Sensors 2021; 21.
https://doi.org/10.3390/s21010198
[21] Ali Almagbile JW, Ding WD. Evaluating the performances of adaptive Kalman filter methods in GPS/INS integration. 2010.
https://doi.org/10.5081/jgps.9.1.33
[22] Li WL etl. Robust unscented Kalman filter with adaptation of process and measurement noise covariances. Digital Signal Processing 2016; 48: 93-103.
https://doi.org/10.1016/j.dsp.2015.09.004
[23] Kontoroupi T and Smyth AW. Online Noise Identification for Joint State and Parameter Estimation of Nonlinear Systems. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering 2016; 2: 1-12.
https://doi.org/10.1061/AJRUA6.0000839
[24] Dalir M. Bashour M. Applications of fractional calculus. Applied Mathematical Sciences 2010; 4: 1021–1032.
[25] Torabi H, Pariz N and Karimpour A. A novel cubature statistically linearized Kalman filter for fractional-order nonlinear discrete-time stochastic systems. Journal of Vibration & Control 2018; 24: 5880-5897.
https://doi.org/10.1177/1077546317692943
[26] Sierociuk D eta. Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation. International Journal of Applied Mathematics and Computer Science 2006; 16: 129-140.
[27] Sun Y, Wang Y, Wu X, et al. Robust extended fractional Kalman filter for nonlinear fractional system with missing measurements. Journal of the Franklin Institute 2018; 355: 361-380.
https://doi.org/10.1016/j.jfranklin.2017.10.030
[28] R. Caballero-Águila AH-CaJL-P. Extended and Unscented Filtering Algorithms in Nonlinear Fractional Order Systems with Uncertain Observations. Applied Mathematical Sciences 2012; 6: 1471-1486.
[29] Sun Y, Wu X, Cao J, Wei Z, Sun G. Fractional extended Kalman filtering for non-linear fractional system with Lévy noises. IET Control Theory & Applications 2017; 11: 349-358.
https://doi.org/10.1049/iet-cta.2016.1041
[30] Gao Z. Reduced order Kalman filter for a continuous-time fractional-order system using fractional-order average derivative. Applied mathematics and computation 2018; 338: 72-86.
https://doi.org/10.1016/j.amc.2018.06.006
[31] Gao Z. Fractional-order Kalman filters for continuous-time fractional-order systems involving colored process and measurement noises. Journal of the Franklin Institute 2018; 355: 922-948.
https://doi.org/10.1016/j.jfranklin.2017.11.037
[32] Chen X, Gao Z, Ma R, Huang X. Hybrid extended-unscented Kalman filters for continuous-time nonlinear fractional-order systems involving process and measurement noises. Transactions of the Institute of Measurement and Control 2020; 42(9): 1618-1631.
https://doi.org/10.1177/0142331219893788
[33] Ramezani Abdolrahman SB. A Modified Fractional-Order Unscented Kalman Filter for Nonlinear Fractional-Order Systems. Circuits, Systems, and Signal Processing 2018; 37: 3756-3784.
https://doi.org/10.1007/s00034-017-0729-9
[34] Kui X, Wentao Y, Feng Q, et al. Adaptive fractional‐order unscented Kalman filter with unknown noise statistics. International Journal of Adaptive Control and Signal Processing 2022; 36: 2519-2536.
https://doi.org/10.1002/acs.3472
[35] Gao Z. Cubature Kalman filters for nonlinear continuous-time fractional-order systems with uncorrelated and correlated noises. Nonlinear Dynamics 2019; 96: 1805–1817.
https://doi.org/10.1007/s11071-019-04885-y
[36] Tianyu Liu SC, Yiheng Wei, Ang Li, Yong Wang. Fractional central difference Kalman filter with unknown prior information. Signal Processing 2019; 154: 294-303. Article.
https://doi.org/10.1016/j.sigpro.2018.08.006
[37] Abdolrahman Ramezani BS, Jafar Zarei. Novel Hybrid Robust Fractional interpolatory Cubature Kalman Filters. Journal of the Franklin Institute 2020; 357: 704–725.
https://doi.org/10.1016/j.jfranklin.2019.11.002
[38] Xiao K, Yu W, Qu F, et al. Adaptive fractional-order unscented Kalman filter with unknown noise statistics. International Journal of Adaptive Control and Signal Process 2022; 36: 2519-2536.
https://doi.org/10.1002/acs.3472
[39] Yang C, Gao Z, Li XN, et al. Adaptive fractional-order Kalman filters for continuous-time nonlinear fractional-order systems with unknown parameters and fractional-orders. International Journal of Systems Science 2021; 52: 2777-2797.
https://doi.org/10.1080/00207721.2021.1904303
[40] Jiang T, Wang J, He Y, et al. Design of the modified fractional central difference Kalman filters under stochastic colored noises. ISA Transactions 2022; 127: 487-500.
https://doi.org/10.1016/j.isatra.2021.08.044
[41] Jiang T and Wang Y. Robust Fractional Nonlinear State Estimation Against Random Incomplete Measurements and Unknown Noise Statistics. IEEE Transactions on Instrumentation and Measurement 2023; 72: 1-11.
https://doi.org/10.1109/TIM.2022.3223147
[42] Mu J, Cheng JL and Xu L. Influence of Fractional order on Performance of Fractional Higher Interpolatory Cubature Kalman Filter. In: 24-th China Conference on System Simulation Technology and its Applications (CCSSTA24th), Hefei China, 2023; p. 5.
[43] Jiang T, Chen J and Wang Y. A novel fractional nonlinear state estimation algorithm in non-Gaussian noise environment. Measurement 2024; 227: 114261.
https://doi.org/10.1016/j.measurement.2024.114261
[44] Genz A and Keister BD. Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. Journal of Computational and Applied Mathematics 1996; 71: 299-309.
https://doi.org/10.1016/0377-0427(95)00232-4
[45] Ienkaran Arasaratnam SH, Thomas R. Hurd. Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations. IEEE Transactions on Signal Processing 2010; 58: 4977-4993.
https://doi.org/10.1109/TSP.2010.2056923
[46] Wang J. Stochastic Modeling for Real-Time Kinematic GPS/GLONASS Position. Navigation 1999; 46.
https://doi.org/10.1002/j.2161-4296.1999.tb02416.x
[47] Mu J and Cai Y. Likelihood-based iteration square-root cubature Kalman filter with applications to state estimation of re-entry ballistic target. Transactions Institute Measusement and Control 2012; 35: 949-958.
https://doi.org/10.1177/0142331212459880
[48] Farina A, Ristic B and Benvenuti D. Tracking a ballistic target: comparison of several nonlinear filters. IEEE Transactions on Aerospace and Electronic Systems 2002; 38: 854-867.
